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11.7 Synchrotron radiation from electrons

The magnetic field not only modifies the rates of neutrino emission processes, but also opens new channels of neutrino emission. Let us consider a coordinate system such that BBB = [0,0,B ]. The electron energy levels are then given by ε (p ) = c(m2c2 + 2ℏω m n + p2)1∕2 n z e c e z, where n is the Landau quantum number. The transverse component of the electron momentum is not conserved, and this allows for the electron synchrotron radiation process,
− − e (BBB ) −→ e (BBB ) + νℓ + ¯νx , ℓ = e,μ,τ , (273 )
where the notation BBB is to remind us of the necessary presence of the magnetic field. For nonquantizing BBB (T ≫ TB) and nondegenerate nonrelativistic electrons, the neutrino emission rate is temperature independent syn 6 Q ν ∝ B. But for a hot (T ≫ TB) ultra-relativistic (2 kBT ≫ mec) electron-positron gas, we get a density independent Qsyνn ∝ B2T 5 (see [428Jump To The Next Citation Point]).
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